[time 924] Re: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof ofunitarity

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 9 Oct 1999 18:15:54 +0300 (EET DST)

On Sat, 9 Oct 1999, Hitoshi Kitada wrote:

> Dear Matti,
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
> Subject: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your
> proof of unitarity
> skip
> > > I am speaking of general context without such a condition. If the limit
> above
> > > exists, then it follows from it the unitarity.
> > >
> > You argue that you can avoid somehow the assumption about the
> > existence of the time development operator and get unitarity from
> > algebraic structure alone. Or that you have unitary time development
> > operator in case that you have only E=0 states
> > of Hamiltonian?

> This is not my argument, but it is an old theory of T. Kato and S. T. Kuroda:

I managed to modify purely formal unitarity proof by starting
from proof which I found from Merzbacher to apply to my case.
It is so horribly formal that it makes me sick(;-).
See the separate posting.

I however tend to believe to the condition V|m_1>=0 since it forces
p-adics and is consistent with the idea of renormalization group
invariance and leads to the cohomological interpretation.
The idea that S-matrix is so near to trivial one that one is forced
to go from reals to p-adics somehow satisfies my pathological mind
(I like dancing on the rope)(;-)-

> Theory of simple scattering and eigenfunction expansions, Functional Analysis
> and Related Topics, Springer-Verlag, 1970, pp. 99-131,

I hope this is in the library.

> and
> The abstract theory of scattering, Rocky Mount. J. Math., Vol. 1 (1971),
> 127-171.
> I am not sure if your interaction term satisfies their assumptions. If it
> works with your case, their argument treats the Hamiltonian without assuming
> conditions like Virasoro conditions. They get a unitarity (completeness in
> their terminology) for general spectra. The spectral projection onto the space
> corresponding to E=0 would then give your unitarity.

I am afraid that I cannot judge whether these assumptions are satisfied.
In any case, it is interesting to look it.

> The problem may be in the interaction term if their method does not work.
> A question related with this is if the E=0 states are genuine eigenvectors
> or generalized ones. Maybe to see if this is the case or not is included in
> your problem?

I am not sure what you mean with generalized eigenvector.


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